Question

Find z 1 z 2 if z 1 = 3(cos37° + isin37°) and z 2 = (cos53° + isin53°

Answer 1

Answer:

$Z_{1} Z_{2} = 3 i$

Step-by-step explanation:

Step(i):-

Given z 1 = 3(cos 37° + i sin 37°)

          z 2 = (cos 53° + i sin 53°)

by using complex numbers

      $Z_{1} = r_{1} ( cos\alpha_{1} + isin\alpha_{1} ) = r_{1} cis\alpha _{1}$

      $Z_{2} = r_{2} ( cos\alpha_{2} + isin\alpha_{2} ) = r_{2} cis\alpha _{2}$

step(ii):-

now

   $Z_{1} Z_{2} = r_{1} r_{2} cis (\alpha _{1} + \alpha _{2})$

       z ₁ = 3(cos 37° + i sin 37°) = 3 c i s 37°

       z ₂ = (cos 53° + i sin 53°) = c i s 53°

we will use formula

     $Z_{1} Z_{2} = r_{1} r_{2} cis (\alpha _{1} + \alpha _{2})$

     $Z_{1} Z_{2} = 3 X 1 cis (37 + 53) = 3cis (90) = 3 cis(\frac{\pi }{2} )$

    $Z_{1} Z_{2} = 3(cos(\frac{\pi }{2} ) + isin(\frac{\pi }{2}))$

    $Z_{1} Z_{2} = 3(0 + i (1)) = 3 i$

Conclusion:-

       $Z_{1} Z_{2} = 3 i$